What is the derivative of (t-sqrt(t))/(t^{1/3)} ?

The derivative of (t-sqrt(t))/(t^{1/3)} is (sqrt(t)-1)/(6t^{5/6)}+1/(2t^{1/3)}

What is the first derivative of (t-sqrt(t))/(t^{1/3)} ?

The first derivative of (t-sqrt(t))/(t^{1/3)} is (sqrt(t)-1)/(6t^{5/6)}+1/(2t^{1/3)}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

derivative of

\frac{t - t}{t ^{\frac{1}{3}}}

\frac{t - 1}{6 t ^{\frac{5}{6}}} + \frac{1}{2 t ^{\frac{1}{3}}}

Solution steps

\frac{d}{d t} (\frac{t - t}{t ^{\frac{1}{3}}})

Simplify

\frac{t - t}{t ^{\frac{1}{3}}} : t^{\frac{1}{6}} (t - 1)

= \frac{d}{d t} (t^{\frac{1}{6}} (t - 1))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = t^{\frac{1}{6}}, g = t - 1

= \frac{d}{d t} (t^{\frac{1}{6}}) (t - 1) + \frac{d}{d t} (t - 1) t^{\frac{1}{6}}

\frac{d}{d t} (t^{\frac{1}{6}}) = \frac{1}{6 t ^{\frac{5}{6}}}

\frac{d}{d t} (t - 1) = \frac{1}{2 t ^{\frac{1}{2}}}

= \frac{1}{6 t ^{\frac{5}{6}}} (t - 1) + \frac{1}{2 t ^{\frac{1}{2}}} t^{\frac{1}{6}}

Simplify

\frac{1}{6 t ^{\frac{5}{6}}} (t - 1) + \frac{1}{2 t ^{\frac{1}{2}}} t^{\frac{1}{6}} : \frac{t - 1}{6 t ^{\frac{5}{6}}} + \frac{1}{2 t ^{\frac{1}{3}}}

= \frac{t - 1}{6 t ^{\frac{5}{6}}} + \frac{1}{2 t ^{\frac{1}{3}}}

What is the derivative of (t-sqrt(t))/(t^{1/3)} ?
The derivative of (t-sqrt(t))/(t^{1/3)} is (sqrt(t)-1)/(6t^{5/6)}+1/(2t^{1/3)}
What is the first derivative of (t-sqrt(t))/(t^{1/3)} ?
The first derivative of (t-sqrt(t))/(t^{1/3)} is (sqrt(t)-1)/(6t^{5/6)}+1/(2t^{1/3)}